Classifications

• • H—ELECTRICITY
  • H04—ELECTRIC COMMUNICATION TECHNIQUE
  • H04L—TRANSMISSION OF DIGITAL INFORMATION, e.g. TELEGRAPHIC COMMUNICATION
  • H04L9/00—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols
  • H04L9/008—Cryptographic mechanisms or cryptographic arrangements for secret or secure communications; Network security protocols involving homomorphic encryption

Definitions

• This invention relates to the field of encryption and, more particularly, to a method useful in securely computing on encrypted data.
• the present invention relates to a method to securely compare two ciphertexts, which are each respective encryptions of two possibly different symbols, to form, without the use of secret keys or decryption, a ciphertext containing an encrypted indication of whether the symbols are the same.
• Homomorphic encryption is a form of encryption which enables the performing of an operation on a pair of ciphertexts, producing a result which when decrypted is the same as if a corresponding operation had been performed on the plaintexts.
• the ciphertext operations for performing homomorphic multiplication and addition are referred to herein as EvalMult and EvalAdd, respectively.
• EvalAdd and EvalMult operations are understood to be modulus-2 operations, i.e., they are modulus-2 homomorphic addition and modulus-2 homomorphic multiplication, respectively.
• a homomorphic encryption scheme is referred to herein as somewhat homomorphic if its homomorphic characteristics support only a finite number of sequential EvalAdd or EvalMult operations.
• the number of EvalMult operations that may be performed on ciphertexts while ensuring that the result, when decrypted, will equal the product of the corresponding plaintexts is referred to herein as the multiplicative degree, or the depth, of the encryption scheme.
• An additive degree may be defined in an analogous manner.
• a somewhat homomorphic encryption scheme may have infinite additive degree but finite multiplicative degree.
• a homomorphic encryption scheme which has infinite additive degree and infinite multiplicative degree is referred to herein as a fully homomorphic encryption scheme.
• An encryption scheme may be referred to as partially homomorphic if it supports only an EvalAdd or an EvalMult operation, but not both.
• Homomorphic encryption may be useful, for example if an untrusted party is charged with processing data without having access to the data.
• a trusted party or data proprietor may encrypt the data, deliver it to the untrusted party, the untrusted party may process the encrypted data and return it to the data proprietor or turn it over to another trusted party. The recipient may then decrypt the results to extract the decrypted, processed data.
• the operations desired may include comparison of symbols.
• An untrusted party may, for example, receive ciphertexts corresponding to two plaintext symbols from one or more data proprietors, and may wish to send a third party an encrypted indication of whether the plaintext symbols are the same, which the third party may decrypt, obtaining for example a binary 1 if the symbols match, i.e., are identical, and a binary 0 if they do not match.
• a binary 1 if the symbols match, i.e., are identical, and a binary 0 if they do not match.
• This invention enables fundamental capabilities for secure computing on encrypted data.
• a user can encrypt data, share the data with an untrusted third party that can compute algorithms on this data without access the original data or encryption keys such that the result of running the algorithm on the encrypted data can be decrypted to a result which is equivalent to the result of running the algorithm on the original unencrypted data.
• a user may encrypt two symbols, share the encrypted symbols with an untrusted third party, and the untrusted third party may perform a sequence of operations on the encrypted symbols to produce an encrypted result which, when decrypted by a trusted party, indicates whether the two symbols are the same.
• This invention could be used by cloud computing hosts, financial institutions and any other commercial entity that may like to use or offer secure computing.
• the secure symbol matching provides the capability to securely determine whether two symbols are equal on unsecure hardware without sharing data in the clear or secret keys.
• This invention requires the use of either a fully encryption (FHE) or a sufficient somewhat homomorphic encryption (SHE) scheme. FHE and SHE schemes support computation on encrypted data.
• a method for comparing a first symbol and a second symbol including: performing a sequence of operations, on: a first set of ciphertexts corresponding to the first symbol; and a second set of ciphertexts corresponding to the second symbol, to form a resulting ciphertext containing an encrypted indication of whether the first symbol matches the second symbol.
• the sequence of operations includes one or more EvalAdd operations and one or more EvalMult operations.
• the method includes decrypting the resulting ciphertext.
• the method includes: converting the first symbol into a binary representation of the first symbol; encoding the binary representation of the first symbol to form a first set of plaintext vectors; encrypting the first set of plaintext vectors with a homomorphic encryption scheme to form the first set of ciphertexts; converting the second symbol into a binary representation of the second symbol; encoding the binary representation of the second symbol to form a second set of plaintext vectors; and encrypting the second set of plaintext vectors with the homomorphic encryption scheme to form the second set of ciphertexts.
• the encoding of the binary representation of the first symbol to form a first set of plaintext vectors includes forming, for each bit of the binary representation, a vector of bits, the bit of the binary representation being the most significant bit of the vector of bits, and the remaining bits of the vector of bits being zero; and the encoding of the binary representation of the second symbol to form a second set of plaintext vectors includes forming, for each bit of the binary representation, a vector of bits, the bit of the binary representation being the most significant bit of the vector of bits, and the remaining bits of the vector of bits being zero.
• the sequence of operations includes: for each first ciphertext of the first set of ciphertexts, and for each corresponding second ciphertext of the second set of ciphertexts: homomorphically adding the first ciphertext and the second ciphertext to obtain a first encrypted sum; homomorphically inverting the first encrypted sum to obtain an encrypted factor, and homomorphically multiplying the encrypted factors to form the resulting ciphertext.
• the homomorphically adding of the first ciphertext and the second ciphertext to obtain a first encrypted sum includes performing an EvalAdd operation on the first ciphertext and the second ciphertext.
• the homomorphically inverting of the first encrypted sum to obtain an encrypted factor includes performing an EvalAdd operation with the encrypted sum and a ciphertext encrypting a vector of bits with a leading 1.
• a system for comparing a first symbol and a second symbol including a processing unit configured to: perform a sequence of operations, on: a first set of ciphertexts corresponding to the first symbol; and a second set of ciphertexts corresponding to the second symbol, to form a resulting ciphertext containing an encrypted indication of whether the first symbol matches the second symbol.
• the sequence of operations includes one or more EvalAdd operations and one or more EvalMult operations.
• the processing unit is configured to decrypt the resulting ciphertext.
• the processing unit is configured to: convert the first symbol into a binary representation of the first symbol; encode the binary representation of the first symbol to form a first set of plaintext vectors; encrypt the first set of plaintext vectors with a homomorphic encryption scheme to form the first set of ciphertexts; convert the second symbol into a binary representation of the second symbol; encode the binary representation of the second symbol to form a second set of plaintext vectors; and encrypt the second set of plaintext vectors with the homomorphic encryption scheme to form the second set of ciphertexts.
• the encoding of the binary representation of the first symbol to form a first set of plaintext vectors includes forming, for each bit of the binary representation, a vector of bits, the bit of the binary representation being the most significant bit of the vector of bits, and the remaining bits of the vector of bits being zero; and the encoding of the binary representation of the second symbol to form a second set of plaintext vectors includes forming, for each bit of the binary representation, a vector of bits, the bit of the binary representation being the most significant bit of the vector of bits, and the remaining bits of the vector of bits being zero.
• the sequence of operations includes: for each first ciphertext of the first set of ciphertexts, and for each corresponding second ciphertext of the second set of ciphertexts: homomorphically adding the first ciphertext and the second ciphertext to obtain a first encrypted sum; homomorphically inverting the first encrypted sum to obtain an encrypted factor, and homomorphically multiplying the encrypted factors to form the resulting ciphertext.
• the homomorphically adding of the first ciphertext and the second ciphertext to obtain a first encrypted sum includes performing an EvalAdd operation on the first ciphertext and the second ciphertext.
• the homomorphically inverting of the first encrypted sum to obtain an encrypted factor includes performing an EvalAdd operation with the encrypted sum and a ciphertext encrypting a vector of bits with a leading 1.
• FIG. 1 is a dataflow diagram illustrating data flows, processing steps, intermediate data products, and a final result in a method for secure symbol matching according to an embodiment of the present invention
• FIG. 2 is a flowchart illustrating a method for secure symbol matching according to an embodiment of the present invention.
• FIG. 3 is a flowchart of a subset of acts executed in a method for secure symbol matching according to an embodiment of the present invention.
• each of two symbols selected from a set of symbols referred to herein as an alphabet, is used to form a sequence of ciphertexts, which are then processed with a sequence of operations to form a resulting ciphertext, the resulting ciphertext containing an encrypted indication of whether the two symbols match, i.e., are the same.
• each symbol is first mapped to a sequence of bits, i.e., a binary representation of the symbol is formed.
• ASCII American Standard Code for Information Interchange
• ASCII maps an alphabet (referred to herein as the ASCII alphabet) including the most common characters in western languages, to a set of 7-bit binary integers.
• Each of the bits bij may then be encoded to a plaintext vector of bits (bij, 0,0,0, ... ,0); ( elements 125, 130 in FIG. 1).
• the plaintext vector is formed by making the bit bij the most significant bit of the plaintext vector, and padding the remainder of the plaintext vector with zeros, to a total length of m bits.
• Such vectors of bits of length m are referred to herein as m-bit- vectors; an m-bit- vector in which the first bit is a 1 is referred to as an m-bit- vector with leading 1, and an m-bit- vector in which the first bit is a 0 is referred to as an m-bit-vector with leading 0.
• the m-bit- vectors are encrypted using a homomorphic encryption scheme to form two sets of ciphertexts 135, 140, one set for each of the symbols, and each ciphertext corresponding to one bit of the binary representation of one symbol.
• This operation is represented symbolically as cij - Enc((bij,0,0,0,...,0)).
• each ciphertext c3j encrypts an m-bit vector with leading 0 if the corresponding bits of the two symbols are the same, and encrypts an m-bit vector with leading 1 if the corresponding bits of the two symbols are different.
• the ciphertext c4j contains an encrypted indication of whether the j* bits of the binary representations of the two plaintext symbols are the same.
• c4j encrypts an m-bit-vector with a leading 1 if the j bits of the binary representations of the two plaintext symbols are the same and c4j encrypts an m-bit-vector with a leading 0 if the j" 1 bits of the binary representations of the two plaintext symbols are not the same.
• ciphertexts c4j are then combined by a sequence of EvalMult operations, to form a ciphertext cfinal, which contains an encrypted indication of whether the two plaintext symbols are the same, i.e., of whether each bit of the binary representation of the first plaintext symbol si is the same as the corresponding bit of the binary representation of the second plaintext symbol s2, for all ⁇ bits.
• a homomorphic logical AND operation is well suited for this purpose, and the EvalMult operation, which implements modulus-2 multiplication, is equivalent to a logical AND of the most significant bits of two vectors, the remaining bits being zeros.
• the product of multiple factors ⁇ 41 * ⁇ 2*...* ⁇ 4 ⁇ employed in the expression for cfinal above may be implemented by operating on the factors and intermediate products pairwise using the EvalMult(a,b) operation until only one final product remains.
• EvalMult(c41,c42,...,c47) may be evaluated as
• C41*c42*c43*c44*c45*c46*c47 (((c41*c42)*(c43*c44))*((c45*c46)*c47))
• (a*b) is used as shorthand notation for EvalMult(a,b)
• the above expression requires, as can be seen by counting the nesting depth of the parentheses, at least a somewhat homomorphic encryption scheme of degree 3.
• a minimum-degree EvalMult operation may be defined recursively using the relation EvalMult(al,a2, ...
• EvalMult(c41,c42,...,c4 ) implements a homomorphic logical AND of the ciphertexts ⁇ 41, ⁇ 42,..., ⁇ 4 ⁇ , each of which encrypts an m-bit- vector with a leading 1 if the corresponding bits of the two symbols match, and an m-bit-vector with a leading 0 if the corresponding bits of the two symbols do not match
• cfinal ⁇ 3 ⁇ ( ⁇ 41, ⁇ 42,..., ⁇ 4 ⁇ ) encrypts an m-bit-vector with a leading 1 if the two symbols match, and an m-bit- vector with a leading 0 if the two symbols do not match.
• decrypting cfinal results in a vector 155 with components (i3, 0, 0, 0), in which i3 is one if the two symbols match, and zero if they do not. Extracting the leading bit 160 from this vector results in a single bit which is 1 if the plaintext symbols match, and 0 if they do not.
• Embodiments of the present invention may be practiced with any SHE (or FHE) scheme configuration as long as it provides adequate degree ceil(log2( ⁇ )) in EvalMult computations, assuming that the encryption scheme has infinite additive degree.
• FIG. 2 summarizes the process according to one embodiment.
• an act 205 two plaintexts are mapped to corresponding ⁇ -bit binary representations.
• each bit of each of the two ⁇ -bit binary representations is used as the most significant bit of an m-bit vector, with the remainder of the vector padded with zeros.
• each vector is encrypted into a corresponding ciphertext, and in act 220, a sequence of EvalAdd and EvalMult operations is performed on the ciphertexts.
• the ciphertext is decrypted, and in an act 230, the leading bit, which is 1 if the plaintext symbols match, and 0 they do not match, is extracted.
• FIG. 3 shows details of the sequence of operations of act 220 (FIG. 2).
• act 305 the ciphertexts are added pairwise using the EvalAdd operation.
• act 310 the sums are inverted, and in an act 315, the ciphertexts resulting from act 310 are all multiplied together homomorphically, by repeated application of the EvalMult operation.
• Processing unit is used herein to include any combination of hardware, firmware, and software, employed to process data or digital signals.
• Processing unit hardware may include, for example, application specific integrated circuits (ASICs), general purpose or special purpose central processing units (CPUs), digital signal processors (DSPs), graphics processing units (GPUs), and programmable logic devices such as field programmable gate arrays (FPGAs).
• ASICs application specific integrated circuits
• CPUs general purpose or special purpose central processing units
• DSPs digital signal processors
• GPUs graphics processing units
• FPGAs field programmable gate arrays

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• Engineering & Computer Science (AREA)
• Computer Security & Cryptography (AREA)
• Computer Networks & Wireless Communication (AREA)
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• Information Retrieval, Db Structures And Fs Structures Therefor (AREA)
• Management, Administration, Business Operations System, And Electronic Commerce (AREA)
• Storage Device Security (AREA)

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• 2013
  • 2013-11-14 EP EP13854202.2A patent/EP2920907A2/de not_active Withdrawn
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  • 2013-11-15 US US14/081,668 patent/US9893880B2/en active Active

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